excursions in modern mathematics, 7e: 10.3 - 2copyright © 2010 pearson education, inc. 10 the...

Excursions in Modern Mathematics, 7e: 10.3 - 2Copyright © 2010 Pearson Education, Inc.

10 The Mathematics of Money

10.1 Percentages

10.2 Simple Interest

10.3 Compound Interest

10.4 Geometric Sequences

10.5 Deferred Annuities: Planned Savings for the Future

10.6 Installment Loans: The Cost of Financing the Present


Under simple interest the gains on an investment are constant–only the principal generates interest. Under compound interest, not only does the original principal generate interest, so does the previously accumulated interest. All other things being equal, money invested under compound interest grows a lot faster than money invested under simple interest, and this difference gets magnified over time. If you are investing for the long haul (a college trust fund, a retirement account, etc.), always look for compound interest.

Compound Interest


Imagine that you have just discovered the following bit of startling news: On the day you were born, your Uncle Nick deposited $5000 in your name in a trust fund that pays a 6% APR. One of the provisions of the trust fund was that you couldn’t touch the money until you turned 18. You are now 18 years, 10 months old and you are wondering, How much money is in the trust fund now? How much money would there be in the trust fund if I waited until my next birthday when I turn 19?

Example 10.10 Your Trust Fund Found!


How much money would there be in the trust fund if I left the money in for retirement and waited until I turned 60?

Here is an abbreviated timeline of the money in your trust fund, starting with the day you were born:


■ Day you were born: Uncle Nick deposits $5000 in trust fund.

■ First birthday: 6% interest is added to the account. Balance in account is (1.06)$5000.


■ Second birthday: 6% interest is added to the previous balance (in red).Balance in account is(1.06)(1.06)$5000 = (1.06)2$5000.

■ Third birthday: 6% interest is added to the previous balance (again in red).Balance in account is(1.06)(1.06)2$5000 = (1.06)3$5000.


At this point you might have noticed that the exponent of (1.06) in the right-hand expression goes up by 1 on each birthday and in fact matches the birthday.


Thus,

■ Eighteenth birthday: The balance in the account is (1.06)18$5000. It is now finally time to pull out a calculator and do the computation:

(1.06)18$5000 = $14,271.70

(rounded to the nearest penny)



■ Today: Since the bank only credits interest to your account once a year and you haven’t turned 19 yet, the balance in the account is still $14,271.70.

■ Nineteenth birthday: The future value of the account is

(1.06)19$5000 = $15,128




Moving further along into the future,

■ 60th birthday: The future value of the account is

(1.06)60$5000 = $164,938.45

which is an amazing return for a $5000 investment (if you are willing to wait, of course)!



This figure plots the growth of the money in the account for the first 18 years.



This figure plots the growth of the money in the account for 60 years.



The future value F of P dollars compounded annually for t years at an APR of R% is given by

F = P(1 + r)t

ANNUAL COMPOUNDING FORMULA


Imagine that you have $875 in savings that you want to invest. Your goal is to have $2000 saved in 7 1/2 years. (You want to send your mom on a cruise on her 50th birthday.) Imagine now that the credit union around the corner offers a certificate of deposit (CD) with an APR of 6 3/4% compounded annually. What is the future value of your $875 in 7 1/2 years? If you are short of your $2000 target, how much more would you need to invest to meet that target?

Example 10.11 Saving for a Cruise


To answer the first question, we just apply the annual compounding formula with P = $875, R = 6.75 (i.e., r = 0.0675), and t = 7 (recall that with annual compounding, fractions of a year don’t count) and get

$875(1.0675)7 = $1382.24




Unfortunately, this is quite a bit short of the $2000 you want to have saved. To determine how much principal to start with to reach a future value target of F = $2000 in 7 years at 6.75% annual interest, we solve for P in terms of F in the annual compounding formula. In this case substituting $2000 for F gives

$2000 = P(1.0675)7



$2000 = P(1.0675)7

and solving for P gives

This is quite a bit more than the $875 you have right now, so this option is not viable. Don’t despair–we’ll explore some other options throughout this chapter.


P $2000

1.0675 7$1266.06


Let’s now return to our story from Example 10.11: You have $875 saved up and a 7 1/2 -year window in which to invest your money. As discussed in Example 10.11, the 6.75% APR compounded annually gives a future value of only $1382.24 – far short of your goal of $2000.

Example 10.12 Saving for a Cruise: Part 2


Now imagine that you find another bank that is advertising a 6.75% APR that is compounded monthly (i.e., the interest is computed and added to the principal at the end of each month). It seems reasonable to expect that the monthly compounding could make a difference and make this a better investment. Moreover, unlike the case of annual compounding, you get interest for that extra half a year at the end.



To do the computation we will have to use a variation of the annual compounding formula. The key observation is that since the interest is compounded 12 times a year, the monthly interest rate is 6.75% ÷ 12 = 0.5625% (0.005625 when written in decimal form). An abbreviated chronology of how the money grows looks something like this:

■ Original deposit: $875.



■ Month 1: 0.5625% interest is added to the account. The balance in the account is now (1.005625)$875.

■ Month 2: 0.5625% interest is added to the previous balance. The balance in the account is now (1.005625)2$875.

■ Month 3: 0.5625% interest is added to the previous balance. The balance in the account is now (1.005625)3$875.



■ Month 12: At the end of the first year the balance in the account is

(1.005625)12$875 = $935.92

After 7 1/2 years, or 90 months,

■ Month 90: The balance in the account is

(1.005625)90$875 = $1449.62



The story continues. Imagine you find a bank that pays a 6.75% APR that is compounded daily. You are excited! This will surely bring you a lot closer to your $2000 goal. Let’s try to compute the future value of $875 in 7 1/2 years. The analysis is the same as in Example 10.12, except now the interest is compounded 365 times a year (never mind leap years–they don’t count in banking), and the numbers are not as nice.



First, we divide the APR of 6.75% by 365.This gives a daily interest rate of6.75% ÷ 365 ≈ 0.01849315% = 0.0001849315

Next, we compute the number of days in the 7 1/2 year life of the investment

365 7.5 = 2737.5

Since parts of days don’t count, we round down to 2737. Thus,

F = (1.0001849315)2737$875 = $1451.47



Let’s summarize the results of Examples

10.11, 10.12, and 10.13. Each example

represents a scenario in which the present

value is P = $875, the APR is 6.75% (r =

0.0675), and the length of the investment is

t = 7 1/2 years. The difference is the

frequency of compounding during the year.

Differences: Compounding Frequency


■ Annual compounding (Example 10.11):

Future value is F = $1382.24.

■ Monthly compounding (Example 10.12):


■ Daily compounding (Example 10.13):




A reasonable conclusion from these numbers is that increasing the frequency of compounding (hourly, every minute, every second, every nanosecond) is not going to increase the ending balance by very much. The explanation for this surprising law of diminishing returns will be given shortly.



The future value of P dollars in t years at an APR of R% compounded n times a year is

GENERAL COMPOUNDING FORMULA

F P 1

r

n

nt


In the general compounding formula, r/n represents the periodic interest rate expressed as a decimal, and the exponent n

• t represents the total number of compounding periods over the life of the investment. If we use p to denote the periodic interest rate and T to denote the total number of times the principal is compounded over the life of the investment, the general compounding formula takes the following particularly nice form.

A Better Looking Form


The future value F of P dollars compounded a total of T times at a periodic interest rate p is

GENERAL COMPOUNDING FORMULA(VERSION 2)

F P 1 p T


One of the remarkable properties of the general compounding formula is that even as n (the frequency of compounding) grows without limit, the future value F approaches a limiting value L. This limiting value represents the future value of an investment under continuous compounding (i.e., the compounding occurs over infinitely short time intervals) and is given by the following continuous compounding formula.

Continuous Compounding


The future value F of P dollars compounded continuously for t years at an APR of R% is

CONTINUOUS COMPOUNDING FORMULA

F Pert


You finally found a bank that offers an APR of 6.75% compounded continuously. Using the continuous compounding formula and a calculator, you find that the future value of your $875 in 7 1/2 years is

F = $875(e7.50.0675)

= $875(e0.50625)

= $1451.68



The most disappointing thing is that when you compare this future value with the future value under daily compounding (Example 10.13), the difference is 21¢.



The annual percentage yield (APY) of an investment (sometimes called the effective rate) is the percentage of profit that the investment generates in a one-year period. For example, if you start with $1000 and after one year you have $1099.60, you have made a profit of $99.60. The $99.60 expressed as a percentage of the $1000 principal is 9.96%–this is your APY.

Annual Percentage Yield


Suppose that you invest $835.25. At the end of a year your money grows to $932.80. (The details of how your money grew to $932.80 are irrelevant for the purposes of our computation.) Here is how you compute the APY:

Example 10.15 Computing an APY

APY

$932.80 $835.25

$835.250.1168 11.68%


In general, if you start with S dollars at the beginning of the year and your investment grows to E dollars by the end of the year, the APY is the ratio (E – S)/S. You may recognize this ratio from Section 10.1–it is the annual percentage increase of your investment.

Annual Percentage Yield


Which of the following three investments is better: (a) 6.7% APR compounded continuously, (b) 6.75% APR compounded monthly, or (c) 6.8% APR compounded quarterly? Notice that the question is independent of the principal P and the length of the investment t. To compare these investments we will compute their APYs.

Example 10.16 Comparing Investments Through APY


(a) The future value of $1 in 1 year at 6.7% interest compounded continuously is given by e0.067 ≈ 1.06930. (Here we used the continuous compounding formula). The APY in this case is 6.93%. (The beauty of using $1 as the principal is that this last computation is trivial.)



(b) The future value of $1 in 1 year at 6.75% interest compounded monthly is (1 + 0.0675/12)12 ≈ 1.00562512 ≈ 1.06963 (Here we used the general compounding formula). The APY in this case is 6.963%.



(b) The future value of $1 in 1 year at 6.8% interest compounded quarterly is(1 + 0.068/4)4 ≈ 1.0174 ≈ 1.06975The APY in this case is 6.975%.



Although they are all quite close, we can now see that (c) is the best choice, (b) is the second-best choice, and (a) is the worst choice. Although the differences between the three investments may appear insignificant when we look at the effect over one year, these differences become quite significant when we invest over longer periods.


excursions in modern mathematics, 7e: 10.3 - 2copyright © 2010 pearson education, inc. 10 the...

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